Problem: Rewrite as a simplified fraction. ${0.\overline{1} = {?}}$
Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}10x &= 1.1111...\\ 1x &= 0.1111...\end{align*}}$ Subtracting the two equations, we have: ${9x = 1}$ ${x = \dfrac{1}{9}} $